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Theorem sxsigon 30771
Description: A product sigma-algebra is a sigma-algebra on the product of the bases. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Assertion
Ref Expression
sxsigon ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))

Proof of Theorem sxsigon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sxsiga 30770 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ ran sigAlgebra)
2 eqid 2799 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
32sxval 30769 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
43unieqd 4638 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
5 mpt2exga 7482 . . . . 5 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
6 rnexg 7332 . . . . 5 ((𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
7 unisg 30722 . . . . 5 (ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
85, 6, 73syl 18 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
94, 8eqtrd 2833 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
10 eqid 2799 . . . 4 𝑆 = 𝑆
11 eqid 2799 . . . 4 𝑇 = 𝑇
122, 10, 11txuni2 21697 . . 3 ( 𝑆 × 𝑇) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
139, 12syl6reqr 2852 . 2 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇))
14 issgon 30702 . 2 ((𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)) ↔ ((𝑆 ×s 𝑇) ∈ ran sigAlgebra ∧ ( 𝑆 × 𝑇) = (𝑆 ×s 𝑇)))
151, 13, 14sylanbrc 579 1 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆 ×s 𝑇) ∈ (sigAlgebra‘( 𝑆 × 𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385   cuni 4628   × cxp 5310  ran crn 5313  cfv 6101  (class class class)co 6878  cmpt2 6880  sigAlgebracsiga 30686  sigaGencsigagen 30717   ×s csx 30767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-siga 30687  df-sigagen 30718  df-sx 30768
This theorem is referenced by:  sxuni  30772
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